T-Tail HALE 03：Linearization 与 Flutter damping¶

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这一页从 trim 点出发，运行 Modal、LinearAssembler 和 AsymptoticStability。除单点特征值外，这里也给出一个轻量扫速，用来展示 flutter 分析里最常看的 damping value、damping trend 和代表性频率变化。

0. 导入公共工具¶

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import os
import sys
from pathlib import Path

os.environ.setdefault("MPLCONFIGDIR", str(Path("_sharpy_runs") / ".mplconfig"))

candidate_dirs = [
    Path.cwd(),
    Path.cwd() / "docs" / "reference-aircraft",
    Path("docs/reference-aircraft"),
]
for candidate in candidate_dirs:
    if (candidate / "ttail_hale_utils.py").exists():
        sys.path.insert(0, str(candidate.resolve()))
        break
else:
    raise FileNotFoundError("没有找到 ttail_hale_utils.py。请从仓库根目录或 docs/reference-aircraft 目录运行本 Notebook。")

import matplotlib.pyplot as plt
import numpy as np
from IPython.display import HTML, display

import ttail_hale_utils as hale

hale.configure_plots()
import os
import sys
from pathlib import Path

os.environ.setdefault("MPLCONFIGDIR", str(Path("_sharpy_runs") / ".mplconfig"))

candidate_dirs = [
    Path.cwd(),
    Path.cwd() / "docs" / "reference-aircraft",
    Path("docs/reference-aircraft"),
]
for candidate in candidate_dirs:
    if (candidate / "ttail_hale_utils.py").exists():
        sys.path.insert(0, str(candidate.resolve()))
        break
else:
    raise FileNotFoundError("没有找到 ttail_hale_utils.py。请从仓库根目录或 docs/reference-aircraft 目录运行本 Notebook。")

import matplotlib.pyplot as plt
import numpy as np
from IPython.display import HTML, display

import ttail_hale_utils as hale

hale.configure_plots()

1. 写入 linearization 配置¶

这里的 flow 是：

BeamLoader -> AerogridLoader -> StaticTrim -> Modal -> LinearAssembler -> AsymptoticStability

num_modes = 12 用来让教学版本更轻；如果要做正式 flutter 边界搜索，可以把模态数、num_evals 和速度点加密。

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case_dir = hale.prepare_hale_case("03_linearization", force=True)

linear_config = hale.write_solver_variant(
    case_dir,
    "ttail_hale_linearization.sharpy",
    ["BeamLoader", "AerogridLoader", "StaticTrim", "Modal", "LinearAssembler", "AsymptoticStability"],
    n_steps=2,
    num_modes=12,
    num_evals=40,
)

print("linearization config:", linear_config)
case_dir = hale.prepare_hale_case("03_linearization", force=True)

linear_config = hale.write_solver_variant(
    case_dir,
    "ttail_hale_linearization.sharpy",
    ["BeamLoader", "AerogridLoader", "StaticTrim", "Modal", "LinearAssembler", "AsymptoticStability"],
    n_steps=2,
    num_modes=12,
    num_evals=40,
)

print("linearization config:", linear_config)

linearization config: /Users/raiot/Documents/Programing/Steps-to-Aeroelastic-Sim/_sharpy_runs/ttail_hale_reference_aircraft/03_linearization/simple_HALE/ttail_hale_linearization.sharpy

2. 运行 linearization¶

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linear_data, linear_log = hale.run_solver(linear_config)
ss = linear_data.linear.ss

print("state-space A shape:", ss.A.shape)
print("inputs:", ss.inputs)
print("outputs:", ss.outputs)
print("captured solver messages:", len(linear_log))
linear_data, linear_log = hale.run_solver(linear_config)
ss = linear_data.linear.ss

print("state-space A shape:", ss.A.shape)
print("inputs:", ss.inputs)
print("outputs:", ss.outputs)
print("captured solver messages:", len(linear_log))

state-space A shape: (2578, 2578)
inputs: 1183
outputs: 1179
captured solver messages: 4

3. 结构模态频率¶

Modal 会把结构模态分析输出到 beam_modal_analysis/eigenvaluetable.txt。这里先看前若干个非零频率，确认线性化前的结构部分不是空结果。

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modal_rows = hale.modal_rows(case_dir)
nonzero_modal = [row for row in modal_rows if row.get("freq_n (Hz)", 0.0) > 1e-6]
display(HTML(hale.html_table(nonzero_modal[:12])))

fig, ax = plt.subplots(figsize=(7.2, 3.2))
ax.bar(
    [int(row["mode"]) for row in nonzero_modal[:12]],
    [row["freq_n (Hz)"] for row in nonzero_modal[:12]],
    color=hale.COLORS["blue"],
)
ax.set_title("First non-zero structural modal frequencies", loc="left")
ax.set_xlabel("mode")
ax.set_ylabel("freq_n [Hz]")
ax.grid(True, axis="y", color=hale.COLORS["light_grey"], linewidth=0.6)
hale.finish_figure(fig)
modal_rows = hale.modal_rows(case_dir)
nonzero_modal = [row for row in modal_rows if row.get("freq_n (Hz)", 0.0) > 1e-6]
display(HTML(hale.html_table(nonzero_modal[:12])))

fig, ax = plt.subplots(figsize=(7.2, 3.2))
ax.bar(
    [int(row["mode"]) for row in nonzero_modal[:12]],
    [row["freq_n (Hz)"] for row in nonzero_modal[:12]],
    color=hale.COLORS["blue"],
)
ax.set_title("First non-zero structural modal frequencies", loc="left")
ax.set_xlabel("mode")
ax.set_ylabel("freq_n [Hz]")
ax.grid(True, axis="y", color=hale.COLORS["light_grey"], linewidth=0.6)
hale.finish_figure(fig)

No rows.

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4. 单点 damping values¶

AsymptoticStability 会把代表性特征值写成表格。这里先把共轭模态折叠成更容易读的一组 damping value，再画特征值位置和 damping-frequency 图。表中的 damping 符号沿用 SHARPy 输出定义，读 flutter 时重点看 damping 是否向零附近移动，同时结合 real(lambda) 判断稳定性。

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stability_rows = hale.stability_rows(case_dir)
representative_modes = hale.representative_stability_modes(stability_rows, max_modes=12)
display(HTML(hale.html_table(
    representative_modes,
    columns=["mode", "eval_real", "eval_imag", "freq_d (Hz)", "damping", "period (s)"],
    precision=5,
)))
hale.plot_stability_rows(stability_rows)
stability_rows = hale.stability_rows(case_dir)
representative_modes = hale.representative_stability_modes(stability_rows, max_modes=12)
display(HTML(hale.html_table(
    representative_modes,
    columns=["mode", "eval_real", "eval_imag", "freq_d (Hz)", "damping", "period (s)"],
    precision=5,
)))
hale.plot_stability_rows(stability_rows)

mode	eval_real	eval_imag	freq_d (Hz)	damping	period (s)
0	34.36697	125.66371	20.00000	-0.26380	0.05000
1	31.51895	125.66371	20.00000	-0.24328	0.05000
2	8.51248	117.65913	18.72603	-0.07216	0.05340
4	8.40623	117.75383	18.74110	-0.07121	0.05336
6	6.89650	-105.41739	16.77770	-0.06528	0.05960
8	6.88002	105.44173	16.78157	-0.06511	0.05959
10	4.72855	113.92873	18.13232	-0.04147	0.05515
12	4.64540	113.91029	18.12938	-0.04075	0.05516
14	0.07084	114.96669	18.29752	-0.00062	0.05465
16	0.00446	-115.00811	18.30411	-0.00004	0.05463
28	-0.05856	-0.55681	0.08862	0.10459	11.28427
30	-0.06485	-0.30718	0.04889	0.20656	20.45430

5. Flutter speed sweep：damping 随速度变化¶

老师说 flutter analysis 要有 damping plot 和 damping value，这里就用两个已确认可稳定收敛的速度点做一个轻量扫速。它不是正式的 flutter boundary 搜索，而是给出完整管线：每个速度重新 trim、linearize，再从稳定性表里抽取最小 damping、最大 real(lambda) 和对应频率。

正式分析时可以把 SWEEP_SPEEDS 加密，例如 np.linspace(6.0, 18.0, 13)，并增加 num_modes 与 num_evals。

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SWEEP_SPEEDS = [8.0, 10.0]

flutter_rows = hale.flutter_speed_sweep(
    SWEEP_SPEEDS,
    label_prefix="03_flutter_damping",
    num_modes=8,
    num_evals=24,
    force=True,
)

display(HTML(hale.html_table(
    flutter_rows,
    columns=[
        "U_inf [m/s]",
        "min damping",
        "mode @ min damping",
        "freq_d @ min damping [Hz]",
        "max real(lambda)",
        "mode @ max real",
        "eigenvalue rows",
    ],
    precision=5,
)))
hale.plot_flutter_sweep(flutter_rows)
SWEEP_SPEEDS = [8.0, 10.0]

flutter_rows = hale.flutter_speed_sweep(
    SWEEP_SPEEDS,
    label_prefix="03_flutter_damping",
    num_modes=8,
    num_evals=24,
    force=True,
)

display(HTML(hale.html_table(
    flutter_rows,
    columns=[
        "U_inf [m/s]",
        "min damping",
        "mode @ min damping",
        "freq_d @ min damping [Hz]",
        "max real(lambda)",
        "mode @ max real",
        "eigenvalue rows",
    ],
    precision=5,
)))
hale.plot_flutter_sweep(flutter_rows)

U_inf [m/s]	min damping	mode @ min damping	freq_d @ min damping [Hz]	max real(lambda)	mode @ max real	eigenvalue rows
8.00000	-0.53895	0	2.61665	10.51930	0	24
10.00000	-0.26380	0	20.00000	34.36697	0	24

6. 线性化点的三维构型¶

linearization 不是围绕原始几何做，而是围绕 trim 后的状态做。下面显示 StaticTrim 之后进入 LinearAssembler 的气动网格。

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traces = hale.aircraft_traces_from_data(linear_data, timestep=-1, opacity=0.42, suffix=" (linearization point)")
hale.show_plotly(
    traces,
    "Linearization point after StaticTrim",
    camera={"eye": {"x": 1.55, "y": -1.9, "z": 0.85}},
)
traces = hale.aircraft_traces_from_data(linear_data, timestep=-1, opacity=0.42, suffix=" (linearization point)")
hale.show_plotly(
    traces,
    "Linearization point after StaticTrim",
    camera={"eye": {"x": 1.55, "y": -1.9, "z": 0.85}},
)